HOMEWORK 7
Exercise 1. Let x : H*(X) <8> H*(Y) —> H*(X x F) be the cross product denned by the formula
otxP=p*1(ot)Up*M where p\ : X x Y —> X and P2 : X x "K —y Y are the projections on the first and the second component, respectively. Let A : X —> X x X be the diagonal A(x) = (x,x). Prove that for a, f3 G
aU/3 = A*(a x /3).
Exercise 2. Let / : X —> Y be a constant map. Prove that /* : Hn(X) —> Hn(Y) and /* : Hn(Y) —> H^X) are zero maps for n > 1. (Hint: One can do it from the definition, but much easier is to factor / as a composition of suitable two maps and use the fact that      and H* are a functor and a cofunctor, respectively.)
Exercise 3. Let the cohomology rings of the spaces X and Y are the following
H*(X) = Z[x]/{xn),   H*(Y) = Z[y]/{ym) where x G and y G Prove that
H*(X V7) =Z[M,w]/(Mn,t;m,Mt;}.
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